Aircraft: EMB ERJ 145 ER/LR
Phase of flight: Descent
Anomaly: Aircraft Equipment Problem Less Severe; Deviation - Track / Heading All Types; Deviation / Discrepancy - Procedural Clearance; Inflight Event / Encounter Weather / Turbulence

I checked in at 5000 ft in IMC with TRACON (the final approach control before tower) and told him we had light rime ice and immediately asked for lower as we simultaneously got the first indication of a problem which was an EICAS caution for our an anti-ice system; 'Wing Anti-Ice Fail' then 'No Ice Anti-Ice On'. A few seconds later this was followed by a 'Master Warning ICE COND-A/I INOP'. It's at this point I made a request to ATC to exit icing conditions which was a requirement of our QRH procedure.The controller told us to make a right to a 090 degree heading and descend to 4000 ft but I misheard and read back a 190 degree heading and descend to 4000 ft. In the cockpit my First Officer (FO) who was the PF corrected me and said we are to fly a 090 degree heading which we flew. Nonetheless; during the turn the controller asked us to fly a 050 heading and asked us if conditions improved. I indicated that they had improved but this was just for a moment as a few moments later I told him that it was worse than our previous heading as the same EICAS messages returned. I asked if we could get our previous east heading again which he gave us and said go due east at 4000 feet. On this heading we no longer received any further alerts on our EICAS regarding this problem. The controller then asked what kind of icing we got at 4000-5000 ft. It's at this point I explained to the controller why we needed to exit icing conditions and that we were getting light rime icing. As we flew east we got no further speed or altitude instructions from ATC until shortly before intercepting the localizer so I told my FO to keep his speed close to 250 kts out of an abundance of caution for icing on our wing. We were ultimately left high on the approach and task saturated while being given several adjustments to speeds in quick succession during the turn to intercept the approach final to Runway XXL. After instructions to slow to 200 kts and slow to 180 kts his next transmission was blocked and then when I asked to say again; we were told to turn to a 200 heading to intercept the localizer. The controller then asked us to expedite the right turn to 200 to intercept the localizer and to descend to 2500 feet. I told my First Officer to increase our turn radius. It's at this point that my FO told me he felt overwhelmed and relinquished aircraft control to me. As I took control and disengaged the autopilot to increase our rate of turn we were then cleared for the approach and told to maintain 2500 ft until established. As I took over the controls at this very task saturated point I overcorrected on the rate of turn and undershot the localizer but caught it just prior to a full scale right deflection and it's at this point the control told us to now fly a 190 heading to join the localizer. I continued to make appropriate corrections and made a stabilize approach and safe visual landing. The aircraft was written up for EICAS indications after arriving at the gate.I believe the stress of dealing with the anti-ice system failures in addition to the late vectors to intercept the localizer at the high speeds and at a higher than normal altitude increased our workload to a task saturation level that affected our performance. Regardless; we are grateful that this did not lead to a more serious outcome. If we were unable to exit icing conditions I would have declared an emergency and requested 3000 feet as I knew that while conditions were broken from 5000-6000 feet only few clouds were reported at 3000 feet. My FO was very new to 121 operations and this also affected his confidence with dealing with the flying of the aircraft at a critical stage of flight. I also believe that the MEL on our aircraft for an INOP Pack #2 may have likely led to the failure of the anti-ice system. Our MEL did not limit the aircraft from flying into icing conditions but after talking to maintenance they believe that this could have been a contributory factor to the system failure. EMB-145 Captain reported track deviations resulted from high workload associated with wing anti-ice failure.